How Does Temperature Change Along a Helical Path?

[jonroberts74]

the temperature at a point in space is T(x,y,z) = x^2+y^2+z^2

and there is a particle traveling along the helix given by

\sigma (t) =(cos(t),sin(t),t)

a) find T'(t)

T&#039;(t) = \frac{\partial T}{\partial x} \frac{dx}{dt} + \frac{\partial T}{\partial y}\frac{dy}{dt}<br /> + \frac{\partial T}{\partial z} \frac{dz}{dt}

= -2cos(t)sin(t) + 2sin(t)cos(t) +2t = 2t

b) find the temperature at time t = \frac{\pi}{2} + 0.01

= cos^2 (t) + sin^2 (t) + t^2

evaluated at the given t

\approx 3.50how does this look?

thanks!

 

[verty]

The last answer doesn't look right, I mean 1 + t^2, $\pi^2 \over 4$ should be about 2.5, not 3.5.

 

[jonroberts74]

The last answer doesn't look right, I mean 1 + t^2, $\pi^2 \over 4$ should be about 2.5, not 3.5.

1+\left( \frac{\pi}{2} + 0.01\right)^2 = 3.49891702681

 

[SammyS]

the temperature at a point in space is T(x,y,z) = x^2+y^2+z^2

and there is a particle traveling along the helix given by

\sigma (t) =(cos(t),sin(t),t)

a) find T&#039;(t)

T&#039;(t) = \frac{\partial T}{\partial x} \frac{dx}{dt} + \frac{\partial T}{\partial y}\frac{dy}{dt}<br /> + \frac{\partial T}{\partial z} \frac{dz}{dt}

= -2cos(t)sin(t) + 2sin(t)cos(t) +2t = 2t

b) find the temperature at time t = \frac{\pi}{2} + 0.01

= cos^2 (t) + sin^2 (t) + t^2

evaluated at the given t

\approx 3.50


how does this look?

thanks!

It looks good !